First Order Logic

1
First Order Logic

• Chapter 8

Some material adopted from notes by Tim Finin
And Andreas Geyer-Schulz
2
First Order (Predicate) Logic (FOL)

• First-order logic is used to model the world in
  terms of
• objects which are things with individual
  identities
• e.g., individual students, lecturers,
  companies, cars ...
• properties of objects that distinguish them from
  other objects
• e.g., mortal, blue, oval, even, large, ...
• classes of objects (often defined by properties)
• e.g., human, mammal, machine, ...
• relations that hold among objects
• e.g., brother of, bigger than, outside, part
  of, has color, occurs after, owns, a member of,
  ...
• functions which are a subset of the relations in
  which there is only one value'' for any given
  input''.
• e.g., father of, best friend, second half, one
  more than ...

3
Syntax of FOL

• Predicates P(x1, ..., xn)
• P predicate name (x1, ..., xn)
  argument list
• A special function with range T, F
• Examples human(x), / x is a human /
• father(x, y) / x is the father of y /
• When all arguments of a predicate is assigned
  values (said to be instantiated), the predicate
  becomes either true or false, i.e., it becomes a
  proposition. Ex. Father(Fred, Joe)
• A predicate, like a membership function, defines
  a set (or a class) of objects
• Terms (arguments of predicates must be terms)
• Constants are terms (e.g., Fred, a, Z, red,
  etc.)
• Variables are terms (e.g., x, y, z, etc.), a
  variable is instantiated when it is assigned a
  constant as its value
• Functions of terms are terms (e.g., f(x, y, z),
  f(x, g(a)), etc.)
• A term is called a ground term if it does not
  involve variables
• Predicates, though special functions, are not
  terms in FOL

4

• Quantifiers
• Universal quantification ? (or forall)
• (?x)P(x) means that P holds for all values of x
  in the domain associated with that variable.
• E.g., (?x) dolphin(x) gt mammal(x)
• (?x) human(x) gt mortal(x)
• Universal quantifiers often used with
  "implication (gt)" to form "rules" about
  properties of a class
• (?x) student(x) gt smart(x) (All students are
  smart)
• Often associated with English words all,
  everyone, always, etc.
• You rarely use universal quantification to make
  blanket statements about every individual in the
  world (because such statement is hardly true)
• (?x)student(x)smart(x) means everyone in the
  world is a student and is smart.

5

• Existential quantification ?
• (?x)P(x) means that P holds for some value(s) of
  x in the domain associated with that variable.
• E.g., (?x) mammal(x) lays-eggs(x)
• (?x) taller(x, Fred)
• (?x) UMBC-Student (x) taller(x,
  Fred)
• Existential quantifiers usually used with
  (and)" to specify a list of properties about an
  individual.
• (?x) student(x) smart(x) (there is a student
  who is smart.)
• A common mistake is to represent this English
  sentence as the FOL sentence
• (?x) student(x) gt smart(x)
• It also holds if there no student exists in the
  domain because
• student(x) gt smart(x) holds for any individual
  who is not a student.
• Often associated with English words someone,
  sometimes, etc.

6
Scopes of quantifiers

• Each quantified variable has its scope
• (?x)human(x) gt (?y) human(y) father(y, x)
• All occurrences of x within the scope of the
  quantified x refer to the same thing.
• Use different variables for different things
• Switching the order of universal quantifiers does
  not change the meaning
• (?x)(?y)P(x,y) (?y)(?x)P(x,y), can write as
  (?x,y)P(x,y)
• Similarly, you can switch the order of
  existential quantifiers.
• (?x)(?y)P(x,y) (?y)(?x)P(x,y)
• Switching the order of universals and existential
  does change meaning
• Everyone likes someone (?x)(?y)likes(x,y)
• Someone is liked by everyone (?y)(?x) likes(x,y)

7
Sentences are built from terms and atoms

• A term (denoting a individual in the world) is a
  constant symbol, a variable symbol, or a function
  of terms.
• An atom (atomic sentence) is a predicate P(x1,
  ..., xn)
• Ground atom all terms in its arguments are
  ground terms (does not involve variables)
• A ground atom has value true or false (like a
  proposition in PL)
• A literal is either an atom or a negation of an
  atom
• A sentence is an atom, or,
• P, P v Q, P Q, P gt Q, P ltgt Q, (P) where P
  and Q are sentences
• If P is a sentence and x is a variable, then
  (?x)P and (?x)P are sentences
• A well-formed formula (wff) is a sentence
  containing no "free" variables. i.e., all
  variables are "bound" by universal or existential
  quantifiers.
• (?x)P(x,y) has x bound as a universally
  quantified variable, but y is free.

8
A BNF for FOL Sentences

• S ltSentencegt
• ltSentencegt ltAtomicSentencegt
• ltSentencegt ltConnectivegt ltSentencegt
• ltQuantifiergt ltVariablegt,... ltSentencegt
  
• ltSentencegt
• "(" ltSentencegt ")"
• ltAtomicSentencegt ltPredicategt "(" ltTermgt, ...
  ")"
• ltTermgt "" ltTermgt
• ltTermgt ltFunctiongt "(" ltTermgt, ... ")"
• ltConstantgt
• ltVariablegt
• ltConnectivegt v gt ltgt
• ltQuantifiergt ? ?
• ltConstantgt "A" "X1" "John" ...
• ltVariablegt "a" "x" "s" ...
• ltPredicategt "Before" "HasColor" "Raining"
  ...
• ltFunctiongt "Mother" "LeftLegOf" ...
• ltLiteralgt ltAutomicSetencegt
  ltAutomicSetencegt

9
Translating English to FOL

• Every gardener likes the sun.
• (?x) gardener(x) gt likes(x,Sun)
• Not Every gardener likes the sun.
• ((?x) gardener(x) gt likes(x,Sun))
• You can fool some of the people all of the time.
• (?x)(?t) person(x) time(t) gt
  can-be-fooled(x,t)
• You can fool all of the people some of the time.
• (?x)(?t) person(x) time(t) gt
  can-be-fooled(x,t)
• (the time people are fooled may be different)
• You can fool all of the people at some time.
• (?t)(?x) person(x) time(t) gt
  can-be-fooled(x,t)
• (all people are fooled at the same time)
• You can not fool all of the people all of the
  time.
• ((?x)(?t) person(x) time(t) gt
  can-be-fooled(x,t))
• Everyone is younger than his father
• (?x) person(x) gt younger(x, father(x))

10

• All purple mushrooms are poisonous.
• (?x) (mushroom(x) purple(x)) gt poisonous(x)
• No purple mushroom is poisonous.
• (?x) purple(x) mushroom(x) poisonous(x)
• (?x) (mushroom(x) purple(x)) gt poisonous(x)
• There are exactly two purple mushrooms.
• (?x)(Ey) mushroom(x) purple(x) mushroom(y)
  purple(y) (xy)
• (?z) (mushroom(z) purple(z)) gt ((xz) v (yz))
  
• Clinton is not tall.
• tall(Clinton)
• X is above Y if X is directly on top of Y or
  there is a pile of one or more other objects
  directly on top of one another starting with X
  and ending with Y.
• (?x)(?y) above(x,y) ltgt (on(x,y) v (?z) (on(x,z)
  above(z,y)))

11
Example A simple genealogy KB by FOL

• Build a small genealogy knowledge base by FOL
  that
• contains facts of immediate family relations
  (spouses, parents, etc.)
• contains definitions of more complex relations
  (ancestors, relatives)
• is able to answer queries about relationships
  between people
• Predicates
• parent(x, y), child (x, y), father(x, y),
  daughter(x, y), etc.
• spouse(x, y), husband(x, y), wife(x,y)
• ancestor(x, y), descendent(x, y)
• relative(x, y)
• Facts
• husband(Joe, Mary), son(Fred, Joe)
• spouse(John, Nancy), male(John), son(Joe, Nancy)
• father(Jack, Nancy), daughter(Linda, Jack)
• daughter(Liz, Linda)
• etc.

12

• Rules for genealogical relations
• (?x,y) parent(x, y) ltgt child (y, x)
• (?x,y) father(x, y) ltgt parent(x, y) male(x)
  (similarly for mother(x, y))
• (?x,y) daughter(x, y) ltgt child(x, y)
  female(x) (similarly for son(x, y))
• (?x,y) husband(x, y) ltgt spouse(x, y) male(x)
  (similarly for wife(x, y))
• (?x,y) spouse(x, y) ltgt spouse(y, x) (spouse
  relation is symmetric)
• (?x,y) parent(x, y) gt ancestor(x, y)
• (?x,y)(?z) parent(x, z) ancestor(z, y) gt
  ancestor(x, y)
• (?x,y) descendent(x, y) ltgt ancestor(y, x)
• (?x,y)(?z) ancestor(z, x) ancestor(z, y) gt
  relative(x, y)
• (related by common ancestry)
• (?x,y) spouse(x, y) gt relative(x, y) (related
  by marriage)
• (?x,y)(?z) relative(x, z) relative(z, y) gt
  relative(x, y) (transitive)
• (?x,y) relative(x, y) gt relative(y, x)
  (symmetric)
• and similarly son(x, y), mother(x, y), wife(x, y)
• Queries (based on facts from the previous page)
• ancestor(Jack, Fred) / the answer is yes /
• relative(Liz, Mary) / the answer is yes
  /
• relative(Nancy, Mathews)

13
Connections between Forall and Exists

• It is not the case that everyone is ... is
  logically equivalent to There is someone who is
  NOT ...
• No one is ... is logically equivalent to All
  people are NOT ...
• We can relate sentences involving forall and
  exists using De Morgans laws
• (?x)P(x) ltgt (?x) P(x)
• (?x) P(x) ltgt (?x) P(x)
• (?x) P(x) ltgt (?x) P(x)
• (?x) P(x) ltgt (?x) P(x)
• Example no one likes everyone
• (?x)(?y)likes(x,y)
• (?x)(?y)likes(x,y)

14
Semantics of FOL

• Domain M the set of all objects in the world (of
  interest)
• Interpretation I includes
• Assign each constant to an object in M
• Define each function of n arguments as a mapping
  Mn gt M
• Define each predicate of n arguments as a mapping
  Mn gt T, F
• Therefore, every ground predicate with any
  instantiation will have a truth value
• In general there are infinite number of
  interpretations because M is infinite
• Define semantics of logical connectives , ,
  v, gt, ltgt as in PL
• Define semantics of (?x) and (?x)
• (?x) P(x) is true iff P(x) is true under all
  interpretations
• (?x) P(x) is true iff P(x) is true under some
  interpretation

15

• Model an interpretation of a set of sentences
  such that every sentence is True
• A sentence is
• satisfiable if it is true under some
  interpretation
• valid if it is true under all possible
  interpretations
• inconsistent if there does not exist any
  interpretation under which the sentence is true
• logical consequence S X if all models of S
  are also models of X

16
Axioms, definitions and theorems

• Axioms are facts and rules which are known (or
  assumed) to be true facts and concepts about a
  domain.
• Mathematicians don't want any unnecessary
  (dependent) axioms -- ones that can be derived
  from other axioms.
• Dependent axioms can make reasoning faster,
  however.
• Choosing a good set of axioms for a domain is a
  kind of design problem.
• A definition of a predicate is of the form P(x)
  ltgt S(x) (define P(x) by S(x)) and can be
  decomposed into two parts
• Necessary description P(x) gt S(x) (only if)
• Sufficient description P(x) lt S(x) (if)
• Some concepts dont have complete definitions
  (e.g. person(x))
• A theorem S is a sentence that logically follows
  the axiom set A, i.e. A S.

17
Higher order logic

• FOL only allows to quantify over variables, and
  variables can only range over objects.
• HOL allows us to quantify over relations
• Example (quantify over functions)
• two functions are equal iff they produce the
  same value for all arguments
• ?f ?g (f g) ltgt (?x f(x) g(x))
• Example (quantify over predicates)
• ?r transitive( r ) gt (?xyz) r(x,y) r(y,z) gt
  r(x,z))
• More expressive, but undecidable.

18
Representing Change

• Representing change in the world in logic can be
  tricky.
• One way is to change the KB
• add and delete sentences from the KB to reflect
  changes.
• How do we remember the past, or reason about
  changes?
• Situation calculus is another way
• A situation is a snapshot of the world at some
  instant in time
• When the agent performs an action A
  in situation S1, the result is a new
  situation S2.

19
Situation Calculus

• A situation is a snapshot of the world at an
  interval of time when nothing changes
• Every true or false statement is made with
  respect to a particular situation.
• Add situation variables to every predicate. E.g.,
  feel(x, hungry) becomes feel(x, hungry, s0) to
  mean that feel(x, hungry) is true in situation
  (i.e., state) s0.
• Or, add a special predicate holds(f,s) that means
  "f is true in situation s. e.g., holds(feel(x,
  hungry), s0)
• Add a new special function called result(a,s)
  that maps current situation s into a new
  situation as a result of performing action a. For
  example, result(eating, s) is a function that
  returns the successor state in which x is no
  longer hungry
• Example The action of eating could be
  represented by
• (?x)(?s)(feel(x, hungry, s) gt feel(x,
  not-hungry,result(eating(x),s))

20
Frame problem

• An action in situation calculus only changes a
  small portion of the current situation
• after eating, x is not-hungry, but many other
  properties related to x (e.g., his height, his
  relations to others such as his parents) are not
  change
• Many other things unrelated to xs feeling are
  not changed
• Explicit copy those unchanged facts/relations
  from the current state to the new state after
  each action is inefficient (and counterintuitive)
• How to represent facts/relations that remain
  unchanged by certain actions is known as frame
  problem, a very tough problem in AI
• One way to address this problem is to add frame
  axioms.
• (?x,s1,s2)P(x, s1)s2result(a(s1)) gtP(x, s2)
• We may need a huge number of frame axioms

21
More on definitions

• A definition of P(x) by S(x)), denoted (?x) P(x)
  ltgt S(x), can be decomposed into two parts
• Necessary description P(x) gt S(x) (only if,
  for P(x) being true, S(x) is necessarily true)
• Sufficient description P(x) lt S(x) (if, S(x)
  being true is sufficient to make P(x) true)
• Examples define father(x, y) by parent(x, y) and
  male(x)
• parent(x, y) is a necessary (but not sufficient )
  description of father(x, y)
• father(x, y) gt parent(x, y), parent(x, y)
  gt father(x, y)
• parent(x, y) male(x) is a necessary and
  sufficient description of father(x, y)
• parent(x, y) male(x) ltgt father(x, y)
• parent(x, y) male(x) age(x, 35) is a
  sufficient (but not necessary) description of
  father(x, y) because
• father(x, y) gt parent(x, y) male(x)
  age(x, 35)

22
More on definitions
S(x) is a necessary condition of P(x)
P(x) S(x)
(?x) P(x) gt S(x)
S(x) is a sufficient condition of P(x)
S(x) P(x)
(?x) P(x) lt S(x)
S(x) is a necessary and sufficient condition of
P(x)
P(x) S(x)
(?x) P(x) ltgt S(x)